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IN  MEMORIAM 
FLOR1AN  CAJOR1 


THE  OUTLOOK 

FOR 

ARITHMETIC 


IN 


AMERICA 


DAVID  EUGENE|  SMITH 

PROFESSOR   OF  MATHEMATICS   IN   TEACHERS   COLLEGE 
COLUMBIA  UNIVERSITY 


PUBLISHERS'  NOTE 

STRANGELY  enough,  arithmetic  is  one  of  the  last 
subjects   in   the   school    curriculum   to   fall   in   line 
with  the  modern  movement  in  education  looking 
to  a   wider   development  and  to  a  more   practical    appli- 
cation.     To  a  great  degree  the  subject  is  still  in  a  state  of 
transition,  and  educators  everywhere  are  deeply  interested 
in  seeing  it  attain  speedily  a  position  of  pedagogical  devel- 
opment commensurate  with  its  importance. 

With  this  thought  in  mind  the  publishers  take  pleasure 
in  presenting  for  the  consideration  of  teachers  the  sugges- 
tions contained  in  this  pamphlet,  which  embody  the  ripest 
thought  of  an  educator  who,  perhaps  more  than  any  one 
else  in  this  country,  has  been  intimately  associated  for 
many  years  with  every  phase  of  mathematical  teaching 
from  the  kindergarten  to  the  university.  As  a  leader  in 
the  pedagogy  of  mathematics,  and  as  the  author  of  a  new 
series  of  mathematical  text-books  for  the  grammar  grades, 
his  comment  will  doubtless  prove  of  value  to  every 
teacher  who  is  interested  in  the  future  development  of  the 
subject. 


GINN  &   COMPANY   PUBLISHERS 

Boston  New  York  Chicago  London 

San  Francisco  Atlanta  Dallas  Columbus 


596-6,  4 


THE  OUTLOOK 

FOR 

ARITHMETIC 

IN 

AMERICA 


DAVID  EUGENE  SMITH 

Professor  of  Mathematics  in  Teachers  College,  Columbia  University.      Author 
of  Smith's  "  Primary  Arithmetic,"  "Grammar  School  Arith- 
metic," and  "Grammar  School  Algebra" 


Copies  of  this  pamphlet  will  be  sent  postpaid  on  request 


CAJOR1 


The  Ideal  Arithmetic 

IN  fine,  a  book  written  for  the  use 
of  those  teachers  who  wish  to  pre- 
serve the  best  that  was  in  the  old- 
style  arithmetic,  with  its  topical  system 
and  its  abundant  drill,  while  giving  to  it 
a  modern  arrangement  and  securing 
"mental  discipline"  through  problems 
of  to-day  rather  than  through  the  tire- 
some, meaningless,  unreal  inheritances 
of  the  past. 

DAVID  EUGENE  SMITH,  in  his 
Preface  to  the  " Grammar  School  Arithmetic" 


The  Outlook  for  Arithmetic 
in  America 

DAVID   EUGENE   SMITH 

Professor  of  Mathematics  in  Teachers  College,  Columbia  University,  New  York  City 


Questions  to  be  considered 

MONG  the  important  question   ever  confronting  those 
teachers  of  arithmetic  who  are  really  leaders,  there  are 
three  or  four  of  peculiar  interest.     They  do  not  disturb 
those  who  are  hopelessly  behind  in  educational  progress, 
nor  those  who,  feeling  that  they  have  attained  perfection, 
wish  the  public  to  know  the  fact.     But  to  the  consider- 
able number  of  earnest  workers  who  are  constantly  seeking  to 
advance  along  safe  lines,  who  have  no  pet  theories  to  exploit, 
and  who  are  open-minded  in  striving  for  the  best,  the  questions 
to  be  considered  are  of  absorbing  interest. 

One  of  these  questions  relates  to  the  immediate  future  of 
arithmetic,  not  in  its  small  details,  but  as  to  its  general  nature. 
Another  inquires  into  the  best  methods  that  have  thus  far  been  sug- 
gested for  teaching  the  subject.  A  third  seeks  to  know  the  probable 
nature  of  the  text-books  that  shall  serve  as  guides.  The  answers  to 
these  questions  are  so  vital  that  their  brief  consideration  in  plain,  non- 
technical language  has  been  made  the  purpose  of  this  monograph. 

Some  of  the  Earlier  Struggles  of  Arithmetic 

Any  investigation  of  the  immediate  future  of  arithmetic  should,  for 
our  present  purposes,  be  limited  to  our  own  country.  For  just  as  that 
teacher  would  be  thought  much  behind  the  times  who  should  teach  litera- 
ture without  emphasizing  the  contributions  of  our  countrymen,  or  history 
without  making  prominent  the  achievements  and  the  responsibilities  of 

3 


Some  of  the  Earlier  Struggles  of  Arithmetic 


our  own  people,  or  geography  without  seeing  in  our  continental  struc- 
ture certain  causes  for  our  varied  national  achievements,  so  the  teacher 
who  presents  arithmetic  without  showing  its  bearing  upon  certain  great 
economic  questions  of  our  own  people  fails  to  keep  abreast  of  the  times 
in  which  he  is  living. 

In  order  the  more  clearly  to  read  the  signs  of  these  times,  we  should 
first  see  how  the  past  has  met  the  conditions  presented  by  succeeding 
generations.  When  the  first  printed  arithmetic  appeared  (at  Treviso, 
Italy,  in  1478),  the  western  business  world  was  just  beginning  to  use 
the  Arabic  numerals.  The  old  Roman  notation  had  been  worthless  for 
computation,  and  people  had  done  their  calculating  with  calculi  (Latin 
for  "pebbles  "),  or  counters,  like  our  checkers,  casting  these  little  disks 
on  a  table,  whence  our  expression  "to  cast  on  account."  Their  accounts 
had  often  been  kept  by  cutting  notches  in  a  stick,  whence  our  expres- 
sion "to  keep  tally"  (French,  tailler,  to  cut).  The  new  arithmetics  tried 
to  break  away  from  such  primitive  plans,  but  it  was  more  than  a  century 
before  they  entirely  succeeded.  Old-style  schoolmasters  talked  loudly 
of  the  arithmetic  of  their  fathers,  of  how  much  better  the  subject  was 
formerly  taught,  of  how  rapidly  they  could  work  in  the  old-fashioned 
way ;  and  they  boasted  of  their  conservatism,  delaying  progress  then 
even  as  it  is  delayed  to-day. 

The  new  arithmetics  sought  also  to  introduce  fresh  problems  of  the 
day,  —  problems  suggested  by  the  awakening  of  commerce  made  possible 
by  Columbus.  But  at  this  the  conservative  element  also  rebelled,  and 
talked  of  the  "  mental  discipline  "  in  the  study  of  amicable  and  deficient 
numbers,  and  of  the  development  of  the  memory  by  learning  definitions 
and  rules  which  the  business  man  never  uses,  and  of  the  acuteness  aris- 
ing from  the  solution  of  problems  about  impossible  men  mowing  imag- 
inary fields  in  unreasonable  hours.  With  this  conservatism  these  early 
books  had  to  struggle ;  but  the  best  ones  succeeded,  and  for  the  next 
century  they  touched  the  actual  life  of  the  people  in  northern  Italy 
and  in  Germany  in  such  a  way  as  to  make  arithmetic  a  vital  subject 
instead  of  a  tradition. 

The  same  battle  had  to  be  fought  in  France  a  little  later  when  she 
became  less  an  agricultural  nation  and  more  given  to  manufacture  and 

4 


The  Immediate  Future  of  Arithmetic  in  America 


to  trade,  and  when  her  educational  system  became  more  independent 
of  the  classical  influence  of  Paris.  England  faced  the  same  problem  a 
little  later  when  she  abandoned  her  position  as  an  agricultural  nation, 
and  Holland  did  the  same  when  she  awoke  to  see  her  advantageous 
position  as  a  maritime  power.  But  in  every  case  the  contest  was  the 
same  :  the  "  mental  discipline  "  of  the  antiquated  chapter  and  the  obso- 
lete problem  are  always  invoked  to  hinder  any  large  view  of  the  demands 
of  the  time. 

The  Immediate  Future  of  Arithmetic  in  America 

To-day  we  are  passing  through  this  same  old  struggle.  America  has 
within  a  generation  awakened  to  a  knowledge  of  her  commanding 
position  as  a  manufacturing  and  trading  nation,  as  distinct  from  an 
agricultural  country  content  with  producing  only  enough  for  her  own 
demands.  Her  position  as  to  arithmetic  is  that  of  Italy  in  1478,  of 
Germany  about  1500,  and  of  France,  England,  and  Holland  about  1575. 
A  demand  is  now  made  that  arithmetic  shall  touch  our  actual  life  in  a 
way  that  it  has  not  in  the  past.  The  schoolmaster  who  has  continued 
to  live  in  the  mental  atmosphere  of  a  generation  ago  may  still  harangue 
about  the  good  old  problems  of  yesterday,  but  our  people,  as  a  whole, 
no  longer  care  about  the  greatest  common  divisor,  cube  root,  such  com- 
mon fractions  as  are  not  needed  in  practical  business ;  about  troy  and 
apothecary's  weight,  compound  numbers  beyond  the  merest  elements, 
compound  proportion,  or,  for  that  matter,  about  simple  proportion 
either.  Alligation,  duodecimals,  equation  of  payments,  and  partner- 
ship involving  time,  have  finally  been  relegated  to  the  arithmetical 
museum,  and  the  good  common  sense  of  our  people  will  demand  that 
these  other  inheritances  follow  them.  This  good  common  sense  will 
tell  them  that  any  "  mental  discipline  "  connected  with  such  topics  can 
as  well  be  secured  from  subjects  that  touch  the  needs  of  the  ordinary 
citizen  to-day.  Because  a  fraction  like  £f  ff  might  have  been  useful 
before  the  world  knew  decimal  fractions,  and  because  the  greatest 
common  divisor  was  necessary  for  its  simplification,  every  one  knows 
to  be  no  sufficient  reasons  for  wasting  a  child's  time  on  such  things 

5 


What  have  the  Method  Writers  Suggested? 


to-day.  And  because  a  lumberman  needs  to  know  Scribner's  rule  and 
a  diamond  merchant  to  know  the  carat  weight  and  a  jeweler  to  know 
the  troy  table  and  a  coal  dealer  the  number  of  tons  to  fill  a  given  bin, 
this  is  no  reason  for  asking  children,  with  no  interest  in  or  prospective 
need  for  these  things,  to  give  any  time  to  them. 

The  great  danger  is  that  we  may  fail  to  supply  the  place  of  the  obso- 
lete matter  with  topics  that  have  an  equal  disciplinary  value.  One  thing 
is  certain,  we  shall  not  succeed  by  simply  putting  in  a  mass  of  algebra 
that  children  and  the  business  world  cannot  use,  and  that  is  as  abstract 
and  unreal  as  what  it  replaces.  Neither  shall  we  succeed  by  hastily 
getting  together  some  work  in  constructive  geometry  with  no  apparent 
end  in  view  except  to  fill  the  gap.  Nor  can  we  hope  to  satisfy  the 
demand,  as  the  present  fashion  seems  to  be,  by  drawing  at  random 
a  body  of  ill-selected  problems  from  science,  by  demanding  an  unrea- 
sonable amount  of  graphic  work  on  uninteresting  statistics,  and  by 
substituting  for  the  topical  method  an  undigested  mass  of  unrelated 
number  facts. 

What  the  new  material  will  probably  be  may  be  inferred  from  the 
experience  of  the  past.  Arithmetic  has  always  failed  of  success  save 
as  it  has  aroused  the  interest  of  children  and  met  the  present  demands 
of  common  life.  To  replace  the  obsolete  matter  by  topics  that  shall  do 
this  is  the  problem  of  the  future. 

What  is  the  Best  that  the  Method  Writers  have 
Suggested  ? 

It  is  not  strange  that  the  word  method  has  been  in  disrepute  with 
the  great  majority  of  the  world's  best  teachers.  The  reason  is  that 
method  has  usually  meant  either  a  hobby,  —  the  magnifying  of  some 
single  idea  to  the  neglect  of  others  of  equal  importance,  —  or  else  it  has 
stood  for  mere  devices  for  doing  something  that  could  as  well  be  done 
in  a  dozen  other  ways.  Of  the  latter,  valuable  as  such  methods  may 
be  in  the  primary  instruction  of  a  teacher,  it  is  not  the  purpose  of  this 
paper  to  speak.  As  to  the  former,  they  are  not  at  all  difficult  to  devise 

6 


What  have  the  Method  Writers  Suggested? 


and  to  exploit.  They  often  make  a  temporary  stir,  and  they  always  sub- 
side thereafter.  In  the  main,  however,  they  do  good;  for  they  make  the 
world  think,  and  they  usually  emphasize,  although  beyond  all  reason, 
some  phase  of  work  that  has  been  neglected. 

Busse,  over  a  century  ago,  suggested  a  valuable  idea  in  his  number 
pictures,  the  basis  of  our  modern  number  cards,  —  an  idea  which  several 
of  his  successors  have  carried  to  such  extremes  as  to  be  ridiculous. 
Pestalozzi,  a  little  later,  gave  the  world  some  exceedingly  valuable  ideas, 
breaking  away  from  the  notion  that  arithmetic  should  not  be  taught 
in  the  first  grade,  and  recognizing  that  a  child  wishes  to  count  as  much 
as  he  wishes  to  read  when  he  comes  to  school.  More  than  any  of  his 
predecessors,  he  asserted  the  necessity  for  knowing  numbers  rather 
than  figures,  of  constant  oral  drill  with  abstract  numbers,  of  using 
simple  material  for  his  early  objective  work,  and  of  abandoning  all 
material  as  soon  as  possible.  A  little  later  Tillich  exploited  a  method 
with  a  set  of  specially  prepared  blocks  and  with  the  ratio  idea  made 
prominent.  With  him  and  with  his  immediate  followers  the  plan  suc- 
ceeded, for  children  adapt  themselves  to  anything ;  you  can  bend  their 
bones,  make  jugglers  of  them,  teach  them  foreign  tongues,  and  make 
them  display  their  powers  in  any  method  that  the  arithmetician  devises. 

About  the  same  time  Kranckes  brought  forward  his  method  of  con- 
centric circles,  proposing  to  teach  a  child  all  about  numbers  to  10,  then 
to  100,  then  to  1000,  and  then  to  10,000,  —  an  idea  which  later  led  into 
the  spiral  method  and  influenced  subsequent  courses  of  study.  Grube 
followed  in  1842  with  a  method  which  insisted  that  all  the  processes 
be  taught  simultaneously,  as  if  they  were  equally  important  or  equally 
difficult ;  but  the  plan  had  too  many  absurdities  of  this  kind  to  attract 
any  large  following.  Later  there  arose  the  counting  method  of  Tank 
and  Knilling,  which  asserts  that  the  world  first  needed  numbers  for 
counting  things,  that  the  ratio  idea  is  a  very  late  development,  that 
counting  is  rhythmical  and  therefore  pleasant,  and  that  the  child,  learn- 
ing somewhat  as  the  world  has  learned,  should  base  his  arithmetic  on 
number  series.  It  argues  that  the  series  2,  4,  6,  8,  etc.,  first  learned 
by  counting  objects  and  then  memorized,  is  merely  the  multiplication 
table  of  two  and  the  addition  table  of  its  multiples.  This  being  true,  it 

7 


The  Probable  Nature  of  our  Future  Arithmetics 


proposes  so  to  organize  counting  exercises  as  to  make  them  the  basis 
for  all  elementary  operations  with  numbers.  There  also  arose  in 
Germany,  about  forty  years  ago,  the  spiral  method,  merely  a  modifica- 
tion of  the  concentric-circle  scheme,  —  an  idea  which  has  influenced 
all  modern  courses  of  study,  but  which  has  gone  to  extremes  that  have 
been  ridiculous.  We  have  also  had  methods  based  on  the  idea  that  a 
number  is  always  an  operator  and  should  always  be  treated  as  such, 
and  we  might  as  well  have  one  asserting  that  a  number  is  an  operand. 
We  have  had  arithmetics  in  rhyme,  schemes  for  basing  all  number  work 
on  the  measurement  of  lines,  others  that  exploited  rectangles,  others 
that  saw  in  paper  folding  the  basis  for  all  primary  arithmetic ;  but  to 
speak  with  any  completeness  of  these  various  plans  is  impossible  in  a 
paper  of  this  kind. 

The  Probable  Nature  of  our  Future  Arithmetics 

In  view  of  the  fact  that  arithmetic  has  never  fulfilled  its  mission 
save  as  it  has  aroused  the  interest  of  the  child  and  as  it  has  touched 
the  real  life  of  the  people,  and  that  no  one  method  has  long  endured, 
although  many  have  served  some  good  purpose,  it  ought  not  to  be  diffi- 
cult to  forecast  the  nature  of  the  arithnvetics  of  the  immediate  future. 

The  primary  text-book  will  probably  be  arranged  for  introduction  in 
the  latter  part  of  the  second  grade  or  early  in  the  third  grade.  The 
children  then  know  enough  about  reading  to  use  a  book  to  advantage, 
and  a  book  serves  to  hold  the  class  and  the  teacher  to  definite,  well- 
planned  work.  It  will  present  the  subject  in  the  sequence  that  the 
experience  of  teachers  (as  set  forth  in  the  best  courses  of  study) 
demands,  and  not  according  to  the  idiosyncrasies  of  the  author.  It 
will  suggest  at  first  a  large  amount  of  thoughtfully  selected,  interesting, 
somewhat  information-giving  oral  work ;  but  it  must  be  recognized 
that  no  text-book  can  ever  be  expected  to  supply  more  than  mere  types 
of  such  work.  As  the  child  progresses,  the  written  work  will  increase, 
and  in  the  grammar  grades  it  will  largely  predominate. 

The  written  work  will  be  elaborate  enough  to  avoid  the  waste  of  time 
attendant  upon  the  dictation  of  problems,  and  it  will  provide  abundant 


The  Probable  Nature  of  our  Future  Arithmetics 


drill  with  abstract  numbers.  The  applied  problems  will  appeal  to  the 
child's  interests,  and  will  not  attempt  to  force  upon  him  ideas  of  busi- 
ness in  advance  of  his  understanding,  or,  what  is  even  worse,  ideas  of 
science  far  beyond  his  years.  They  will  early  touch  upon  his  games, 
his  toys,  his  genuine  purchases,  and  such  simple  handiwork  as  may  be 
within  his  knowledge.  They  may  soon  begin  to  appeal  to  the  heroic, 
being  so  grouped  as  to  tell  such  stories  as  those  of  the  life  savers  on 
our  coasts  and  the  fire  fighters  in  our  towns.  They  will  not  elaborate 
small  details  at  first,  but  will  tell  of  the  large  things  of  nature  in  which 
the  children  take  interest,  of  the  big  things  in  the  sources  of  our 
supply  of  food  and  clothing,  of  the  transportation  of  these  supplies  by 
rail  or  in  ships,  and  of  their  sale  in  the  cities.  But  all  this  will  be 
presented  naturally,  as  applications  of  the  operations  with  numbers 
arranged  in  a  conservative,  steady  progress.  This  progress  will  be 
made  by  adopting  the  best  in  all  the  methods,  but  by  going  to  the 
extreme  in  no  one  of  them.  It  will  be  made  by  using  objects  not 
of  any  one  type  exclusively  but  of  such  various  kinds  as  to  show  the 
wide  range  of  material  for  illustration.  It  will,  however,  abandon  such 
objective  work  as  soon  as  it  ceases  to  be  necessary,  the  crutch  being 
harmful  as  soon  as  the  invalid  is  ready  to  develop  his  own  strength. 

The  book  of  the  future  will  not  go  to  the  extreme  of  measuring  any 
and  every  thing  with  no  well-defined  purpose  in  view,  or  of  thinking  that 
its  problems  become  concrete  simply  by  talking  of  butterflies  and  of  fric- 
tion. On  the  contrary  it  will  seek  for  problems  with  a  motive,  and  will 
treat  of  business,  of  science,  and  of  statistics  only  as  these  can  be  made 
real  and  interesting  to  a  child.  A.nd  in  thus  telling  these  stories,  and 
in  showing  the  real  world  on  the  quantitative  side,  the  new  arithmetic 
will  not  fail  to  enter  the  child's  life  in  a  way  that  its  predecessor  did  not. 
By  letting  the  old-style  isolated  problems  about  yards  of  cloth  and 
heads  of  cattle  give  place  to  examples  so  grouped  as  to  impart  real 
information  and  arouse  genuine  interest,  the  new  arithmetic  will  become 
in  the  best  sense  concrete,  as  the  old  one  was  too  often  in  the  worst 
way  abstract. 

And  the  grammar-school  arithmetic,  what  of  that?  It,  too,  will 
probably  follow  the  best  courses  of  study  that  have  been  devised,  and 

9 


The  Probable  Nature  of  our  Future  Arithmetics 


will  thus  assure  a  sequence  that  is  not  eccentric.  As  far  as  circum- 
stances will  permit,  it  will  eliminate  everything  that  is  not  genuinely  use- 
ful, seeking  its  "  mental  discipline  "  through  the  problems  of  real  life. 
For  some  time  to  come  it  will  have  to  cater  enough  to  the  demands  of 
the  past  generation  to  give  work  on  proportion  as  distinct  from  equations, 
on  square  root,  and,  to  a  greater  extent  than  is  really  justifiable,  on  com- 
pound numbers  and  unused  fractions.  But  in  the  applications  of  the 
processes  it  will  so  group  its  problems  as  to  tell  those  parts  of  the  story 
of  our  national  resources  and  business  life  that  are  interesting  and 
intelligible  to  children.  It  will  tell  of  the  manufacturing  of  the  East,  of 
the  cotton  fields  of  the  South,  of  the  mines  of  the  Appalachians  and  the 
Rockies  and  the  Lake  region,  of  the  fruit  industry  of  the  Pacific  coast, 
and  of  the  interests  of  the  great  prairie  country.  It  will  tell  of  the 
transportation  of  the  products  of  the  land,  of  their  manufacture  and 
their  marketing.  It  will  seek  to  ennoble  labor,  to  make  our  national 
economy  interesting,  and  to  prepare  the  boy  and  the  girl  for  the  life 
they  are  soon  to  enter.  It  will  forecast  for  them  the  way  in  which  they 
will  take  their  first  steps  into  business,  will  tell  them  of  their  wages, 
their  bank  accounts,  the  common  commercial  papers  they  will  use,  and 
of  the  simple  accounts  of  the  home. 

And  yet  the  new  arithmetic  will  not  be  such  a  great  departure  from 
the  old  in  those  features  that  made  the  old  book  successful,  in  its  way. 
It  will  not  offer  an  aimless  mass  of  applications,  ill-arranged  and  poorly 
paged.  On  the  contrary,  it  will  keep  to  the  old  topical  arrangement,  so 
that  a  child  may  see  his  own  progress  and  may  have  the  pleasure  aris- 
ing from  feeling  that  he  has  mastered  a  definite  part  of  some  subject. 
It  will,  however,  touch  upon  every  important  topic  at  least  twice,  both 
to  avoid  the  weariness  of  dwelling  too  long  upon  it  and  to  afford  oppor- 
tunity for  a  thorough  review. 

As  to  algebra,  it  will  not  yield  to  the  thoughtless  demand  for  it 
merely  because  it  is  a  higher  subject,  but  it  will  use  the  letter  x  where 
it  is  needed  for  arithmetic,  and  nowhere  else.  If  it  goes  beyond  this, 
it  will  let  algebra  grow  naturally  out  of  arithmetic  and  will  give  as 
many  genuine  applications  as  possible,  as  distinguished  from  the  mere 
puzzles  of  our  common  books.  That  algebra  can  be  taught  to  children 

10 


The  Probable  Nature  of  our  Future  Arithmetics 


will  be  no  argument  for  so  teaching  it,  and  that  by  teaching  it  to  a  child 
he  becomes  a  better  arithmetician  has  been  abundantly  disproved  by 
modern  psychology.  If  it  is  taught,  as  it  doubtless  will  be,  it  will  be 
for  other  reasons  than  these. 

Such  books  as  those  described  are  very  hard  to  make.  They  cannot 
be  written  with  the  shears.  They  come  into  being  only  as  the  result  of 
much  thought  and  experiment.  As  no  one  man  has  the  monopoly  of 
the  common  sense  that  inspires  such  books,  so  no  one  person  has  a 
monopoly  of  the  brains  that  will  work  them  out.  The  encouragement  of 
progressive  teachers  will  hasten  their  advent,  just  as  the  carping  of  advo- 
cates for  antiquated  topics  will  retard  it.  But  when  they  really  begin  to 
appear,  there  will  come  a  healthy  reaction  against  the  neglect  of  arith- 
metic that  has,  with  sad  results,  characterized  our  school  work  for  a 
number  of  years  past.  Of  late  we  have  been  letting  arithmetic  slip 
from  our  hands  because  we  could  not  justify  the  general  arrangement 
of  the  books  so  at  variance  with  our  courses  of  study,  or  the  general 
nature  of  the  problems  so  at  variance  with  our  present  life.  That  better 
computers  were  produced  forty  years  ago  than  now  was  not  owing  to  the 
dull  rules,  the  stupid  principles,  and  the  absurd  problems  of  that  day ; 
but  because  the  work  was  definite,  the  teacher  and  the  pupils  could  get 
their  bearings,  the  curriculum  was  confined  to  the  "Three  R's,"  and  this 
particular  "  R  "  had  its  share  of  the  time.  Make  the  new  arithmetic  as 
definite  as  the  old,  preserve  the  best  of  the  topical  arrangement,  give 
enough  problems  and  let  them  appeal  to  the  common-sense  business 
man  as  genuine  and  to  his  child  as  interesting,  and  the  arithmetic  of 
the  future  will  be  the  best  that  the  world  has  ever  seen. 

Columbia  University,  July  i,  1904. 


ii 


SMITH'S  ARITHMETICS 

PRIMARY    ARITHMETIC 

1 2 mo.      Cloth.      2 64  pages.      Illustrated.      List  price,  30  cents 


~  PRIMARY- 
ARITHMETIC 


WITHOUT  going  to  an  extreme  in  any  one  theory,  this 
book  presents  the  best  modern  ideas  of  primary  arith- 
metic, and  is  intended  to  assist  in  vitalizing  the  work  in  the 
elementary  grades.  In  sequence  of  topics  it  follows  the  best 
city  and  state  courses  of  study  for  the  first  four  school  years. 
While  intended  for  introduction  in  the  latter  half  of  the 
second  year,  and  suggesting  a  scheme  by  half  years,  the  plan 
is  so  elastic  as  to  suit  local  conditions. 

In  the  selection  of  problems,  those  against  which  teachers 
have  long  protested  have  been  replaced  by  those  appealing 
to  the  life,  the  interests,  the  needs,  and  the  powers  of 
children. 

The  book  is  illustrated  with  great  care,  the  pictures  show- 
ing the  relation  of  numbers  and  their  uses  in  measurements,  suggesting  simple  and 
inexpensive  material  for  teachers,  and  rendering  more  interesting  and  real  the  various 
groups  of  problems. 

GRAMMAR    SCHOOL    ARITHMETIC 

I2mo.      Cloth.     394.  pages.      Illustrated.      List  price,  6$  cents 

LIKE  the  author's  "  Primary  Arithmetic,"  this  work  follows,  in  sequence  of  topics, 
the  best  of  the  courses  of  study  in  use  by  the  various  cities  and  states.     In  the 
matter  of  methods  and  device,  the  effort  is  made  to  adopt  the  best,  always  avoiding 
extremes. 

The  exercises  are  composed  of  genuine  American  problems,  showing  common 
business  as  it  really  is  to-day,  but  without  presenting  subjects  too  technical  for  the 
appreciation  of  boys  and  girls  of  grammar-school  age. 

The  pictures  selected  will  appeal  to  teachers  as  reasonable,  and  will  add  to  the 
interest  and  value  of  the  book  for  children. 

IN    PREPARATION 

GRAMMAR  SCHOOL  ALGEBRA 

THIS  work  is  intended  to  serve  as  an  introduction  to  the  study  of  algebra,  and  is 
adapted  to  the  needs  of  the  seventh  or  eighth  school  year.     It  is  arranged  in 
harmony  with  the  leading  courses  of  study  that  include  algebra  in  the  curriculum 
of  the  grades. 

GINN    &    COMPANY    PUBLISHERS 

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